Applications or uses of the Serre-Swan theorem

Solution 1:

One application of this is that topological K-theory (i.e., the K-theory of the exact category of vector bundles on the space) is the same thing as the "algebraic" $K_0$ of the ring of continuous functions (i.e., the K-theory of the exact category of finitely generated projective modules over that ring). So topological K-theory is a "special case" of algebraic K-theory (though the tools for proving things like Bott periodicity are very different from those used in proving general results on exact category in algebraic K-theory).

By the way, here is the corresponding result in algebra:

Algebraic vector bundles over an affine scheme $\mathrm{Spec} A$ are the same as finitely generated projective $A$-modules (let's say $A$ is noetherian). So a module is projective if and only if it is locally free, in algebra language.

Solution 2:

You could perhaps do worse than consulting $\S 6.4$ of my commutative algebra notes: "Applications of Swan's Theorem." (You could definitely do better: see below.)

The first application I give is to show that the ring of real-valued continuous functions on $[0,1]$ is a connected ring in which each finitely generated projective module is free but for which there is a nonfree infinitely generated projective module. As I admit myself in the notes, it is possible to prove this purely algebraically and I allude to another proof taken from one of Lam's books, but the topological approach is a nice one.

The second application is a big one: it exhibits stably free non-free modules over the ring $\mathbb{R}[x_0,\ldots,x_n]/(x_0^2+\ldots + x_n^2 - 1)$ of polynomial functions on the $n$-sphere when $n \neq 0,1,3,7$. This is done by reducing to the known behavior of tangent bundles to the $n$-sphere in the usual differential topological setting.

By the way, I got this second application directly from Swan's paper. There are other applications given there as well...